HMMT 2021 G10

Acute triangle $A B C$ has circumcircle $\Gamma$. Let $M$ be the midpoint of $B C$. Points $P$ and $Q$ lie on $\Gamma$ so that $\angle A P M=90^{\circ}$ and $Q \neq A$ lies on line $A M$. Segments $P Q$ and $B C$ intersect at $S$. Suppose that $B S=1, C S=3, P Q=8 \sqrt{\frac{7}{37}}$, and the radius of $\Gamma$ is $r$. If the sum of all possible values of $r^{2}$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.